The Annals of Probability

Disorder chaos in the Sherrington–Kirkpatrick model with external field

Wei-Kuo Chen

Full-text: Open access

Abstract

We consider a spin system obtained by coupling two distinct Sherrington–Kirkpatrick (SK) models with the same temperature and external field whose Hamiltonians are correlated. The disorder chaos conjecture for the SK model states that the overlap under the corresponding Gibbs measure is essentially concentrated at a single value. In the absence of external field, this statement was first confirmed by Chatterjee [Disorder chaos and multiple valleys in spin glasses (2009) Preprint]. In the present paper, using Guerra’s replica symmetry breaking bound, we prove that the SK model is also chaotic in the presence of the external field and the position of the overlap is determined by an equation related to Guerra’s bound and the Parisi measure.

Article information

Source
Ann. Probab., Volume 41, Number 5 (2013), 3345-3391.

Dates
First available in Project Euclid: 12 September 2013

Permanent link to this document
https://projecteuclid.org/euclid.aop/1378991842

Digital Object Identifier
doi:10.1214/12-AOP793

Mathematical Reviews number (MathSciNet)
MR3127885

Zentralblatt MATH identifier
1303.60089

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)

Keywords
Disorder chaos Guerra’s replica symmetry breaking bound Parisi formula Parisi measure Sherrington–Kirkpatrick model

Citation

Chen, Wei-Kuo. Disorder chaos in the Sherrington–Kirkpatrick model with external field. Ann. Probab. 41 (2013), no. 5, 3345--3391. doi:10.1214/12-AOP793. https://projecteuclid.org/euclid.aop/1378991842


Export citation

References

  • [1] Aizenman, M., Sims, R. and Starr, S. (2003). An extended variational principle for the SK spin-glass model. Phys. Rev. B 68 214403.
  • [2] Bolthausen, E. and Sznitman, A. S. (1998). On Ruelle’s probability cascades and an abstract cavity method. Comm. Math. Phys. 197 247–276.
  • [3] Brary, A. J. and Moore, M. A. (1987). Chaotic nature of the spin-glass phase. Phys. Rev. Lett. 58 57–60.
  • [4] Chatterjee, S. (2009). Disorder chaos and multiple valleys in spin glasses. Preprint. Available at arXiv:0907.3381.
  • [5] Derrida, B. (1981). Random-energy model: An exactly solvable model of disordered systems. Phys. Rev. B (3) 24 2613–2626.
  • [6] Fisher, D. S. and Huse, D. A. (1986). Ordered phase of short range Ising spin glasses. Phys. Rev. Lett. 56 1601–1604.
  • [7] Guerra, F. (2003). Broken replica symmetry bounds in the mean field spin glass model. Comm. Math. Phys. 233 1–12.
  • [8] Katzgraber, H. G. and Krza̧kała, F. (2007). Temperature and disorder chaos in three-dimensional Ising spin glasses. Phys. Rev. Lett. 98 017201.
  • [9] McKay, S. R., Berker, A. N. and Kirkpatrick, S. (1982). Spin-glass behavior in frustrated Ising models with chaotic renormalization-group trajectories. Phys. Rev. Lett. 48 767–770.
  • [10] Panchenko, D. and Talagrand, M. (2007). On the overlap in the multiple spherical SK models. Ann. Probab. 35 2321–2355.
  • [11] Ruelle, D. (1987). A mathematical reformulation of Derrida’s REM and GREM. Comm. Math. Phys. 108 225–239.
  • [12] Sherrington, D. and Kirkpatrick, S. (1975). Solvable model of a spin glass. Phys. Rev. Lett. 35 1792–1796.
  • [13] Talagrand, M. (2006). The Parisi formula. Ann. of Math. (2) 163 221–263.
  • [14] Talagrand, M. (2006). Parisi measures. J. Funct. Anal. 231 269–286.
  • [15] Talagrand, M. (2007). Mean field models for spin glasses: Some obnoxious problems. In Spin Glasses. Lecture Notes in Math. 1900 63–80. Springer, Berlin.
  • [16] Talagrand, M. (2011). Mean Field Models for Spin Glasses. Volume I: Basic Examples. Ergebnisse der Mathematik und Ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics] 54. Springer, Berlin.
  • [17] Talagrand, M. (2011). Mean Field Models for Spin Glasses. Volume II: Advanced Replica-Symmetry and Low Temperature. Ergebnisse der Mathematik und Ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics] 55. Springer, Berlin.